LONDON, Jan 9 (Reuters) - Talk of Britain drastically reworking trade ties with the European Union after Brexit sent the pound tumbling to two-month lows on Monday, as signals that U.S. interest rates could rise three times this year lifted the dollar.

The pound slid over 1 percent against both the dollar and the euro after weekend comments from British Prime Minister Theresa May that she was not interested in keeping “bits of membership” of the European Union.

May said she instead wanted a bespoke deal and also denied criticism that she was “muddled” in the pursuit of what she called the right relationship with the EU, the country’s largest trading partner.

Sterling slid as low as $1.2125, its weakest against the dollar since the end of October. It fell 1.1 percent against the euro too, hitting 86.65 pence per euro, the lowest since mid-November.

“We think it was the prime minister prioritizing immigration over a trade deal that pulled the run from under sterling,” said National Australia Bank’s head of markets strategy, Nick Parsons.

“The optimism in late November, early December had been based on an apparent desire to secure a trade deal and yesterday was a marked shift (from) that,” he added, saying the next threshold to watch for the pound was $1.2083.

The dollar crept ahead after signs of wage pressure in the December U.S. jobs report on Friday proved enough to lift 10-year Treasury yields from 2.33 percent to 2.42 percent after a sizable fall earlier in the week.

Chicago Federal Reserve President Charles Evans, who is a voting Fed member this year, also lent support, saying U.S interest rates could go up three times this year, faster than he expected a few months ago.

The dollar index, which measures the greenback against six top currencies, was up 0.2 percent at 102.43 ahead of U.S. share trading in which the Dow Jones was again eyeing 20,000 points.

TRUMP FACTOR

The euro initially held up after Germany reported the steepest monthly rise in exports in four-and-half years, but the resistance faded and it sagged a few cents to $1.0518 having ricocheted between $1.0339 and $1.0621 last week.

“If we have seen the low in U.S. Treasury yields then potentially the dollar could come back quite strongly,” said Saxo Bank’s head of FX strategy John Hardy.

“It all depends on Trump and how aggressive he is in his first days in office.”

The president-elect is due to be sworn in on Jan. 20 but this week there will be Senate confirmation hearings for a number of his top administration nominees, including secretary of state and attorney general, that should provide a test of his ability to work with fellow Republicans in Congress.

Two non-voting Fed presidents will speak later on Monday, and five speeches are lined up for Thursday. The main economic release of the week is not until Friday, when retail sales figures for December are out.

Dealers in Asia will be keeping an eye on the yuan after Beijing engineered a sharp tightening in liquidity last week that squeezed speculators out of short yuan/long dollar positions.

China’s central bank kept up the pressure on Monday, setting a firmer fix for the yuan than many had expected at 6.9262 per dollar, even though that was down from the previous fix.

Yet the defence is proving costly.

Figures out over the weekend showed China’s foreign exchange reserves fell to nearly six-year lows in December as Beijing fought to stem an outflow of capital that could ultimately force another devaluation of the currency.

For Reuters new Live Markets blog on European and UK stock markets see reuters://realtime/verb=Open/url=http://emea1.apps.cp.extranet.thomsonreuters.biz/cms/?pageId=livemarkets

Additional reporting by Wayne Cole in Sydney; Editing by Toby
Chopra and John Stonestreet